Just regarding the post title, John Birch died 14 years before the JBS was founded and had nothing to do with them. They considered him a martyr because he was killed by a Chinese Communist during an assignment so they used him as a symbol of anti-Communism. James Doolittle, a wartime pilot and possibly Birch's friend (maybe they just worked together, not sure exactly and I don't feel like looking into it further), said in his autobiography:

"[Birch] had no way of knowing that the John Birch Society, a highly vocal postwar anti-communist organization, would be named after him because its founders believed him to be the 'first casualty of World War III.' I feel sure he would not have approved."

As usual it's these far right assholes hiding behind something/someone more respectable to launder their bullshit.